Pyragas-type feedback control for chatter mitigation in high-speed milling

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1 Preprints 2th IFAC Workshop on Time Delay Systems June Ann Arbor MI USA Pyragas-type feedback control for chatter mitigation in high-speed milling N. van de Wouw N.J.M. van Dijk H. Nijmeijer Department Mechanical engineering Eindhoven University of Technology Eindhoven Netherlands Philips Innovation Services Eindhoven Netherlands Abstract: Chatter is an instability phenomenon in high-speed milling that limits machining productivity by the induction of tool vibrations. In this paper a design methodology for loworder Pyragas-type delayed feedback controllers is proposed. These controllers enable dedicated shaping of the chatter stability boundary such that working points of higher machining productivity become feasible while avoiding chatter. The control design problem is cast into a nonsmooth optimization problem which is solved using bundle methods. Distinct benefits of this approach are the a priori fixing of the controller order the limitation of the control action and the fact that no finite-dimensional model approximations and online chatter estimation techniques are required. A representative example illustrates the merit of the proposed methodology in terms of increasing the chatter-free depth of cut thereby enabling significant increases in the productivity of milling processes. Keywords: Robust control delay systems high-speed milling. INTRODUCTION High-speed milling is a widely used manufacturing technique to produce for example moulds and dies or components for the aerospace industry. The productivity in milling is often limited by the occurrence of an instability phenomenon called regenerative) chatter. Chatter causes heavy vibrations of the tool resulting in rapid tool wear an inferior workpiece surface quality and noise. The occurrence of chatter can be analyzed using so-called stability lobes diagrams SLD). In an SLD the chatter stability boundary between a stable cut i.e. without chatter) and an unstable cut i.e. with chatter) is depicted in terms of two key machining parameters: the spindle speed and depth of cut. To overcome chatter vibrations in high-speed milling and therewith to enable the chatterfree increase of the material removal rate dedicated active control strategies are required. Most of the results on active chatter control in milling involve the active damping of the machine dynamics Dohner et al. 2004; Kern et al. 2006) or workpiece Zhang and Sims 2005). Damping the machine or workpiece dynamics either passively or actively results in a uniform increase of the stability boundary for all spindle speeds. To enable more dedicated shaping of the stability boundary e.g. lifting the SLD locally around a specific spindle speed) the regenerative effect which is inherent to the metal cutting process and which is the root cause for N. van de Wouw is also affiliated with the Department of Civil Environmental & Geo- Engineering University of Minnesota Minneapolis U.S.A. and with the Delft Center for Systems and Control Delft University of Technology Delft Netherlands chatter should be taken into account in chatter controller design Shiraishi et al. 99; Chen and Knospe 2007; van Dijk et al. 20a). All aforementioned research either does not include the regenerative effect during controller design or utilizes high-order finite-dimensional approximations of the milling model for controller design yielding high-order controllers which is disadvantageous from an implementation perspective. Building upon the results of van Dijk et al. 20b) this paper presents a low-order controller design methodology which can guarantee chatter-free milling operations in an a priori defined range of process parameters while explicitly taking into account the regenerative effect responsible for chatter. In particular we propose a design for low-order chatter controllers for the milling process using Pyragastype feedback Pyragas 992). The choice for Pyragastype feedback design is motivated as follows. An important aspect of the chatter controller design is the selection of the variable used for feedback see van Dijk et al. 20a). In the latter paper it has been shown that socalled perturbation feedback i.e. only using the chatter vibrations as opposed to the full vibration of the milling machine in the feedback loop ) is favorable in terms of limiting the control action which is important in practice. To enable perturbation feedback online estimation techniques for the chatter vibrations are needed since these cannot be measured directly see van Dijk et al a). Here we propose an alternative way to achieve such perturbation feedback while still only using measurements of the full vibrations of the milling machine. To this end we propose to employ Pyragas -type delayed output feedback which avoids the need for complex online Copyright IFAC

2 TDS 205 June Ann Arbor MI USA estimation techniques for chatter vibrations by the grace of the structure of such type of control. The main contributions of this paper are summarized below. Firstly we propose a fixed-order controller design technique for the high-speed milling process. This technique guarantees the avoidance of chatter in a predefined range of working points in terms of spindle speed and depth-of-cut). In this way large increases in productivity material removal rate) can be achieved while avoiding chatter. Secondly the proposed control strategy has favorable properties from an implementation perspective in three ways. Firstly it allows the user to prespecify the order of the controller and hence supports loworder controller design which is desirable in a real-time implementation. Secondly the proposed strategy limits the control action by employing so-called perturbation feedback and thirdly it implements such perturbation feedback through Pyragas-type delayed output-feedback. Additional contributions of the current paper with respect to van Dijk et al. 20b) are the following: firstly the synthesis methodology has been extended to accommodate the design of Pyragas-type delayed output feedback control and secondly the inclusion of the design of dynamic output feedback chatter controllers which can improve performance with respect to static controllers). 2. HIGH-SPEED MILLING PROCESS A model of the milling process will be described concisely below see e.g.altintas 2000; Faassen et al. 2003; Stépán 200; Insperger et al. 2003) for more details. In Figure a schematic representation of the milling process is depicted. The predefined motion of the tool with respect to the workpiece is characterized in terms of the static chip thickness h jstat t) = f z sinφ j t) where f z is the feed per tooth and φ j t) the rotation angle of the j-th tooth of the tool with respect to the y normal) axis see Figure ). However the total chip thickness h j t) also depends on the interaction between the cutter and the workpiece. This interaction causes cutter vibrations resulting in a dynamic displacement v t t) = v tx t)v ty t)] T of the tool see Figure which is superimposed on the predefined tool motion and results in a wavy workpiece surface. The next tooth encounters Workpiece f z z y x c x b x n Mill c y v ty φ j t) F r f z +v tx Fig.. Schematic representation of the milling process. b y F t this wavy surface generated by the previous tooth and in turn generates its own waviness. This is called the regenerative effect. The difference between the current and previous wavy surface is called the dynamic chip thickness denoted by h jdyn t) = sinφ j t)cosφ j t)]v t t) v t t τ)) with τ = 60/zn) the delay z the number of teeth and n the spindle speed in revolutions per minute rpm). Hence thetotalchipthicknessremovedbytoothj attimeth j t) equals the sum of the static and dynamic chip thickness: h j t) = h jstat t)+h jdyn t). The cutting force model relates the total chip thickness to the forces acting at the tool tip. The tangential and radial forces F t and F r in Figure for a single tooth j are described by the following exponential cutting force model: F tj t) = g j φj t) ) K t a p h j t) xf F rj t) = g j φj t) ) ) K r a p h j t) xf where 0 < x F and K t K r > 0 are cutting parameters which depend on the workpiece material. Moreover a p is the axial depth of cut. The function g j φj t) ) in ) describes whether a tooth is in or out of cut: g j φj t) ) { φs φ j t) φ e h j t) > 0 = 2) 0 else where φ s and φ e are the entry and exit angle of the cut respectively. The total cutting forces in the x- and y- directions F t t) = F tx t) F ty t)] T can be obtained by summing over all z teeth: z F t t) = a p g j φj t) ) h jstat t) with +sinφ j t) cosφ j t)] v t t) v t t τ) )) x FSj t) S j t) = ] cosφj t) sinφ j t). sinφ j t) cosφ j t) Kt K r ] ) 3) The cutting force interacts with the machine spindle and tool) dynamics which are modeled with a linear multiinput-multi-output MIMO) state-space model ẋt) = Axt)+B t F t t)+b a F a t) v t t) = C t xt) v a t) = C a xt) where xt) is the state. F a t) = F ax t) F ay t)] T denote thecontrolforceswheref ax t)andf ay t) arethe control forces acting in the x- and y-direction respectively. 4) Substitution of 3) into 4) results in the nonlinear nonautonomous delay differential equations DDE) describing the dynamics of the milling process: ẋt) = Axt)+B a F a t) z +B t a p g j φj t) ) h jstat t) +sinφ j t) cosφ j t)]c t xt) xt τ) ) ) x FSt) Kt K r ]) v a t) = C a xt). 5) 335

3 TDS 205 June Ann Arbor MI USA The static chip thickness h jstat t) is periodic with period time τ = 60 zn. In general the uncontrolled i.e. F a t) 0) milling model 5) has a periodic solution x t) with period time τ Faassen et al. 2007). In the absence of chatter this periodic solution is locally) asymptotically stable and when chatter occurs it is unstable. Hence the chatter stability boundary can be analyzed by studying the local) stability properties of the periodic solution x t). Hereto the milling model is linearized about the periodic solution x t) for zero control input i.e. F a t) 0) yielding the following linearized dynamics in terms of perturbations xt) xt) = x t)+ xt)): z xt) = A xt)+a p B t H j φ j t))c t xt) xt τ)) +B a F a t) ṽ a t) = C a xt) where H j φ j t)) = g j φj t) ) x F f z sinφ j t)) xf St) Kt 6) ] ] T sinφj t). K r cosφ j t) 7) The linearized model 6) 7) is a delayed periodically time-varying system. As described in Altintas 2000) for full immersion cutsstudied in this paper) it is sufficient to average the dynamic cutting forces z H jφ j t)) over the tool path such that the milling model becomes a timeinvariant DDE model. Since the cutter is only cutting when φ s φ φ e the averaged cutting forces are given by H = z φe z H j φ)dφ. 8) 2π φ s Then a linear time-invariant model of the milling process is obtained by combining 6) with z H jφ j t)) = H and H given in 8). 3. PROBLEM FORMULATION The objective of this paper is to design a low-order linear controller K to generate control inputs F a based on measurements v a which guarantees: robust stability of x = 0 in 6) 7) for uncertainties in depth of cut a p and time delay τ; performance by minimizing the total amount of actuator energy. By ensuring robust stability for uncertainties in a p and τ chatter-free milling operations can be guaranteed in an a priori defined range of spindle speeds n = 60 zτ and depth of cut a p. Moreover we will include the limitation of the actuator forces as a performance criterion in the controller design since it is also an important practical performance specification. An important aspect in the development of an active chatter control design strategy is the selection of the variable used for feedback van Dijk et al. 20a). In particular it is concluded that perturbation feedback i.e. using ṽ a as an input to the controller) is beneficial for reducing required actuator forces without compromising performance in terms of the achievable closed-loop depth q r F a Fig. 2. Generalized plant interconnection. P K of cut/spindle speed interval for which chatter can be eliminated). Here we introduce dynamic Pyragas-type delayed output feedback for robust stabilization of the high-speed milling process to implement such perturbation feedback. The proposed Pyragas-type) controller K with input ṽ a R 2 and output control action) F a R 2 has the following state-space description: ξt) = A c ξt)+b c ṽ a t) ṽ a t τ)) 9) F a t) = C c ξt)+d c ṽ a t) ṽ a t τ)). Herein ξ R nc A c R nc nc B c R nc 2 C c R 2 nc and D c R 2 2 with n c the order of the controller. The benefit of employing Pyragas-type feedback control lies in thefact thatthe signalṽ a t) cantypicallynotbemeasured directly. By realizing that ṽ a t) ṽ a t τ) = v a t) v a t τ) since v at) = v at τ) with v a = C a x ) holds due to the periodic nature of the chatter-free solution the controller in 9) can be directly implemented using only measurement of v a t) without the need for online estimation algorithms to estimate the chatter vibrations ṽ a t)). 4. GENERALIZED PLANT FORMULATION To solve the problem formulated in Section 3 the milling model will be extended with uncertainties in depth of cut a p and spindle speed n. For this purpose the control goal will be cast into the generalized plant framework see Figure 2. The generalized plant P is a given system with three sets of inputs and three sets of outputs. The signal pair p q denote the in-/outputs of the uncertainty channel connecting the plant to the uncertainty block. The signal r represents an external input in which possible disturbances measurement noise and reference inputs are stacked. The signal F a is the control input. The output z can be considered as a performance variable while yt) = C a xt) xt τ)) = ṽ a t) ṽ a t τ) denotes the outputs used for feedback. To construct the generalized plant formulation consider the linear time-invariant model of the milling process obtained by combining 6) with z H jφ j t)) = H and H given in 8). Let us define the following uncertainty sets: a p = 2āp+δ ap ) and τ = τ 0 +δ τ 0) where ā p is the maximal depth of cut for which stable milling is desired δ ap C δ ap τ 0 = τ+τ 2 and δ τ τ τ 2 ] such that 0 < τ < τ. Here ] τ and τ together 60 define the range of spindle speeds z τ 60 zτ for which stable milling is desired. Moreover as motivated in Section 3 it is desired to limit the magnitude of the actuator forces. p z y 336

4 TDS 205 June Ann Arbor MI USA Therefore the performance output z is chosen as the weightedcontrolinputzs) = W KS s)f a s)s Cwhere W KS is a stable weighting filter with the following statespace realization: ẋ KS t) = A KS x KS t)+b KS F a t) ) zt) = C KS x KS t)+d KS F a t). Substituting 0) in 6) with z H jφ j t)) = H and H given in 8) and by adding the performance in-/output channels to the system and rearranging terms) the statespace representation of the generalized plant P is given by: ẋ P t) = A P0 x P t)+a P x P t τ 0 )+B P u P t) 2) v P t) = C P0 x P t)+c P x P t τ 0 )+D P u P t) with the state vector x P t) = x T t) x T KS t)]t input vector u P t) = q T t) r T t) F T at)] T with r R 2 representing measurement noise on the output y and output vector min v P t) = p T t) z T t) y T t)] T. Thestructured) uncertainty sup ) µ N K ω R 20) channel input pt) and output qt) are defined as subject to ΨK) < 0 p t) C t xt τ 0 ) with N the lower fractional transformation LFT) of P in pt) = p 2 t) := C a xt τ 0 ) 9) and fixed-structure controller K in 9) and ΨK) the p 3 t) 2āpC t xt) xt τ 0 )) 2āpq t) spectral abscissa function of the closed-loop system defined 3) as: q t) D δτ )p t) ΨK) := sup{rλ) : detλi Ā 0 Ā e λτ0 ) = 0} 2) qt) = q 2 t) := D δτ )p 2 t) 4) where ] ] ] ] q 3 t) δa p p 3 t) A+ Ā 0 = 2āpB t HCt 0 Ba 0 Dc C + c Ca I B where the delay-operatord δτ isdefined asd δτ xt) = xt c A c 0 I ] ] ] ] δ τ ). The state-space matrices of the generalized plant in Ā 2) are given by = 2āpB t HC t 0 Ba 0 Dc C + c Ca I B c A c 0 0 ] ] A+ A P0 = 2āpB t HC t 0 A 0 A P = 2āpB t HC t 0 The constraint on the objective function defined above is KS 0 0 a necessary condition to guarantee the existence of the ] B P = 2āpB H t H 0 Bt H 0 -norm of N along with stability of the closed-loop Ba system Zhou et al. 996) B KS 0 0 C t C a 0 C P0 = 2āpC t 0 0 C C P = 2āpC t 0 KS 0 0 C a 0 C a D P = 2āpI D KS 0 I 2 0 I 2 0 with identity matrix I n R n n. The transfer function description of the generalized plant P is given by: Ps) = C P0 +C P e sτ0) si A P0 5) A P e sτ0] BP +D P 6) where s C. By s) we denote the Laplace transform of the uncertainty term 4) such that qs) = s)ps) with ] e sδ τ )I s) = ) 0 δ ap I 2 The uncertainty term depends on the frequency. The delay uncertainty e sδτ can be upper bounded by a non-rational) frequency-dependent upper bound κω) as follows Huang and Zhou 2000): 2sin δ τω κω) = 2 ω 0 ω π/δ τ 2 ω π/δ τ 8) in the sense that e iωδτ κω). By defining Ls) = diagκω)i 4 I 2 ) for all s = ξ + iω with ξ ω R the scaled) generalized plant and uncertainty term can be written as follows: Ps) = diagls)i 2 I 2 )Ps) = s)l s). 9) 5. SYNTHESIS OF LOW-ORDER PYRAGAS-TYPE FEEDBACK CONTROLLERS Based on the above clarifies that the design of a controller guaranteeing robustness properties with respect to the scaled) uncertainty requires the solution of the following optimization problem: The robust stability and performance requirement can now be translated into the demand that the objective function in 20) is smaller than one. It is in generalhard to calculateµ N). Neverthelessan upper bound on µ N) can be obtained by calculating the scaled H norm of N Zhou et al. 996). Since the uncertainties are modeled as complex uncertainties see 0) and 7) D-K-iteration see Zhou et al. 996) pro ides a reasonable approach towards solving the problem. Hereby the optimization problem is described by: min inf DND K D H 22) subject to ΨK) < 0 which is iteratively solved for K and D. Herein DND = sup σ Diω)Niω)Diω) ) ω R H denotes the set of functions that are analytic and bounded in the open right half plane and the structure of D is chosen such that D commutes with the uncertainty set i.e. satisfies D = D. For more details on the computation of lower and upper bounds on the complex structured singular value we refer to Packard and Doyle 993). For a given K the problem of finding the scaling matrix D can be formulated as a convex optimization problem which is generally solved pointwise in the frequency domainfor example by using the mussv command 337

5 TDS 205 June Ann Arbor MI USA Table. Milling model parameters. Parameter Value Parameter Value m tx = m ty 0.05 kg K t 462 N/mm +xf) ] m ax = m ay 0.4 kg K r 38.6 N/mm +xf) ] ω tx = ω ty 2350 Hz z 4 -] ω ax = ω ay 400 Hz φ s 0 rad] ζ tx = ζ ty ] φ e π rad] ζ ax = ζ ay 0.2 -] f z 0.2 mm/tooth from the Robust Control Toolbox of Matlab. Given our goal of designing fixed-structure controllers the problem of finding K for a given D in general results in the following non-convex non-smooth constrained optimization problem: minfk) subject to ΨK) < 0 23) K with fk) := sup σ Diω)Niω)Diω) ). ω R The non-smooth dependency of the objective function 23) on the controller parameters of K typically occurs when the maximum of the objective function is located at two or more) different frequencies. Due to the non-smoothness in 23) standard optimization algorithms cannot be used to determine the optimal) parameters of controller K. Instead we employ a non-smooth optimization technique namely a gradient bundle method called gradient sampling developed in Burke et al. 2005). For further details on the optimization-based algorithm and the roleofgradientsamplingherein) used tosolvethe above fixed-structure robust controller synthesis problem and other fixed-order control designs than those based on Pyragas-type feedback we refer to van Dijk 20; van Dijk et al. 205). 6. CONTROLLER SYNTHESIS RESULTS This section presents the results of the application of the controller synthesis methodology presented in Section 5 to the robust chatter control problem. Here the machine spindle-toolholder-tool dynamics in 4) is modeled by two decoupled subsystems representing the dynamics in two xy) orthogonal directions perpendicular to the spindle axis). The dynamics in both the x- and y-directions are modeled as two-degree-of-freedom massspring-damper systems with masses m ik i {at} k {xy} with m tx m ty the tool mass in x- y-direction and m ax m ay the spindle/actuator mass in x- y- direction the eigenfrequencies ω ik = c ik /m ik ) i {at} k {xy}and dimensionlessdamping ratiosζ ik = b ik /2 c ik m ik ) i {at} k {xy}. This model is adopted to capture the inherent dynamics between the actuator/sensor system denoted by subscript a) and the cutting tool denoted by subscript t). The parameters of the milling model are given in Table. Moreover x F = reflecting a nonlinear cutting model. Next the results of synthesizing dynamic Pyragas-type delayed output controllers as defined by 9) are presented. The Pyragas-type dynamic output feedback controller will be designed such that milling operations between n ] rpm are stabilized for a depth of cut which is as large as possible given the performance requirement Table 2. Results from fixed-structure controller synthesis for three different controller orders. n c -] No. D-K-steps µ -] ā p mm] a pmax Depth of cut mm] n c = 4 n c = 2 n c = 0 Without control Domain to be stabilized Spindle speed krpm] Fig. 3. Stability lobes diagram for controllers of order n c = 02 and 4 respectively and without control. The area for which robust stability is guaranteed is indicated by the dashed boxes see also Table 2. on the weighted control sensitivity. Here the performance weighting W KS is chosen as 2πf W KS s) = K rl s+ p 2πf pl s+ 2πf rh s+ 2πf ph s+ 24) with K p = 0 6 mm/n f rl = 00 Hz f rh = 7500 Hz f pl = 0 2 Hz and f ph = Hz. Three low-order Pyragas-type controllers are synthesized namely for n c = 0 i.e. static delayed output feedback) and n c = 2 and n c = 4 i.e. dynamic delayed output feedback) using the algorithm as presented in Section 5. The results are listed in Table 2. Herein ā p denotes the maximal depth of cut for which robust performance can be guaranteed and a pmax denotes the maximal depth of cut in the Stability Lobes Diagram SLD) for the desired spindle speed interval. The resulting controllers are given in Figure 4. This figure shows that the controllers designed for n c = 2 and n c = 4 are dynamic MIMO controllers with notch characteristics. The SLDs are computed using the semi-discretization method Insperger and Stépán 2004) with the designed controllers and without control using the linearized time-variant model of the milling process in 6). The corresponding results are given in Figure 3. From the figure it can be observed that for the case where n c = 0 the fixed-structure controller indeed alters the SLD. In this case based on the SLD the depth of cut can be increased from a pmax =.067 mm in open loop to.469 mm in closed loop which is an improvement of approximately 38%. The peak of the closed-loop stability lobe is approximately located at n = rpm which is outside the domain of desired spindle speeds. Of course if the peak of the SLD could be placed inside the desired interval of spindle speeds then a higher maximum depth of cut couldbe achieved.in orderto shift the peak ofthe lobe at such a spindle speed the controller in this case needs 338

6 TDS 205 June Ann Arbor MI USA Mag. N/mm] Mag. N/mm] Frequency Hz] W KS Frequency Hz] Fig. 4. Magnitude of the fixed-structure controllers of order n c = 0 black dashed) n c = 2 grey solid) and n c = 4 black solid). Also the magnitude of the inverse of the performance weighting function W KS is given. to exhibit more complex dynamics which is obtained by increasing the controller order. For the dynamic fixed-structure controllers with n c = 2 and n c = 4 it can be observed in Figure 3 that the SLD is altered such that a lobe is indeed created at the desired spindle speed interval. Clearly by increasing the order of the fixed-structure controller the area for which robust stability is guaranteed is increased. In this case based on the SLD in Figure 3 the depth of cut can be increased from a pmax =.067 mm to a pmax = 2.46 for n c = 2 and to a pmax = for n c = 4 which leads to a productivity increase of approximately 0% and 25% respectively. Figure 3 clearly illustrates the benefit of dynamic controllers over the static controllers proposed in van Dijk et al. 20b). 7. CONCLUSION This paper presents a strategy for the design of low-order controllers guaranteeing robust stability and performance of the high-speed milling process; in particular the avoidance of chatter in a predefined area of depth-of-cut and spindle speed while respecting limitations regarding the required actuator forces. We have proposed Pyragas-type delayed dynamic outputfeedback controllers thereby simplifying their implementation in practice as no additional estimators for chatterrelated vibrations are needed. Moreover the approach enables the design of relatively low-order controllers which is desirable from a real-time implementation perspective especially given the high-frequency characteristics of the milling dynamics. The presented example illustrates the merit of the proposed controller synthesis strategy in terms of ensuring a significantly higher material removal rate in closed loop while avoiding chatter. REFERENCES Altintas Y. 2000). Manufacturing automation. Cambridge University Press Cambridge UK. Burke J. Lewis A. and Overton M. 2005). A robust gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM Journal on Optimization 53) Chen M. and Knospe C.R. 2007). Control approaches to the suppression of machining chatter using active magnetic bearings. IEEE Trans. on Control Systems Technology 52) Dohner J.L. Lauffer J.P. Hinnerichs T.D. Shankar N. Regelbrugge M.E. Kwan C.M. Xu R. Winterbauer B. and Bridger K. 2004). Mitigation of chatter instabilities in milling by active structural control. Journal of Sound and Vibration 269-2) Faassen R.P.H. van de Wouw N. Nijmeijer H. and Oosterling J.A.J. 2007). 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